Multiplicity of solutions to the generalized extensible beam equations with critical growth

Ground State Solution for a Fourth Order Elliptic Equation of Kirchhoff Type with Critical Growth in ℝ N

“…Motivated by the works we mentioned above, especially by [4,10], we consider the combination of equations ( 7) and ( 11) and extend to the general convolution case in ℝ N . In our paper, we get the ground state solution of problem (1).…”

“…where a > 0, b ≥ 0, λ is a positive parameter, 5 ≤ N ≤ 8, V : ℝ N ⟶ ℝ is a potential function, and I α is a Riesz potential of order α ∈ ðN − 2, NÞ defined by I α = ðΓððN − αÞ/2ÞÞ/ðΓðα/ 2Þπ N/2 2 α jxj N−α Þ. Here, 2 * * = 2N/ðN − 4Þ with N ≥ 5 is the Sobolev critical exponent, and Δ 2 u = ΔðΔuÞ is the biharmonic operator, that is, Δ 2 u = ∑ N i=1 ð∂ 4 /∂x 4 i Þu + ∑ N i≠j ð∂ 4 /∂x 2 i ∂x 2 j Þu. Besides, VðxÞ: ℝ N ⟶ ℝ is a potential function satisfying (V1) V ∈ Cðℝ N , ℝÞ and inf x∈ℝ N VðxÞ ≔ V 0 > 0 (V2) measfx ∈ ℝ N : VðxÞ ≤ Mg < ∞, where meas denotes the Lebesegue measure in ℝ N and V 0 and M are positive constants.…”

“…and λ is a positive parameter. Soon after that, Liang et al [4] obtained multiplicity of solutions to the following generalized extensible beam equation with critical growth:…”